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Unformatted text preview: Engineering 101 Engineering 101 Lecture 27 Lecture 27 Matrices Matrices Prof. Michael Falk University of Michigan, College of Engineering Matrices Matrices One very important data structure for doing engineering and scientific computation is the matrix. A matrix is a two dimensional array. A matrix can be represented as a vector of vectors. 1 2 5 2 4 1 0 2 -1 Matrices Matrices We can think of labeling each element of the matrix by its row and column A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 A ij = Matrices Matrices The numbers of rows and columns can be different from each other A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 A ij = Matrices and Vectors Matrices and Vectors Matrices and vectors can be multiplied by each other. To do this the number of columns in the matrix must equal the number of elements in the vector. A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 b 1 b 2 b 3 b 1 b 2 b 3 Matrices and Vectors Matrices and Vectors To perform the multiplication each row of the matrix is multiplied by the vector and all the results are added up. A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 = A 11 b 1 + A 12 b 2 + A 1 3 b 3 Matrices and Vectors Matrices and Vectors To perform the multiplication each row of the matrix is multiplied by the vector and all the results are added up. A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 = b 1 b 2 b 3 A 11 b 1 + A 12 b 2 + A 1 3 b 3 A 21 b 1 + A 22 b 2 + A 2 3 b 3 Matrices and Vectors Matrices and Vectors To perform the multiplication each row of the matrix is multiplied by the vector and all the results are added up. A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 = b 1 b 2 b 3 A 11 b 1 + A 12 b 2 + A 1 3 b 3 A 21 b 1 + A 22 b 2 + A 2 3 b 3 A 31 b 1 + A 32 b 2 + A 3 Matrices and Vectors Matrices and Vectors To perform the multiplication each row of the matrix is multiplied by the vector and all the results are added up. A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 = A 11 b 1 + A 12 b 2 + A 1 3 b 3 A 21 b 1 + A 22 b 2 + A 2 3 b 3 A 31 b 1 + A 32 b 2 + A 3 b 1 b 2 b 3 Matrices and Vectors Matrices and Vectors To perform the multiplication each row of the matrix is multiplied by the vector and all the results are added up. 4-1 3 7 0 1 1 1 3 2 1 = 4* 3 + 0* 2 + (-1)* 1 3* 3 + 7* 2 + 0 * 1 0* 3 + 1* 2 + 0* 1 1* 3 + 0* 2 + 1* 1 Matrices and Vectors Matrices and Vectors To perform the multiplication each row of the matrix is multiplied by the vector and all the results are added up. 4-1 3 7 0 1 1 1 3 2 1 = 12 + 0 1 9 + 14 + 0 0 + 2 + 0 3 + 0 + 1 Matrices and Vectors Matrices and Vectors To perform the multiplication each row of the matrix is multiplied by the vector and all the results are added up....
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